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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxc8 9801 Extend class notation to include the number 8.

Syntaxc9 9802 Extend class notation to include the number 9.

Syntaxc10 9803 Extend class notation to include the number 10.

Definitiondf-2 9804 Define the number 2. (Contributed by NM, 27-May-1999.)

Definitiondf-3 9805 Define the number 3. (Contributed by NM, 27-May-1999.)

Definitiondf-4 9806 Define the number 4. (Contributed by NM, 27-May-1999.)

Definitiondf-5 9807 Define the number 5. (Contributed by NM, 27-May-1999.)

Definitiondf-6 9808 Define the number 6. (Contributed by NM, 27-May-1999.)

Definitiondf-7 9809 Define the number 7. (Contributed by NM, 27-May-1999.)

Definitiondf-8 9810 Define the number 8. (Contributed by NM, 27-May-1999.)

Definitiondf-9 9811 Define the number 9. (Contributed by NM, 27-May-1999.)

Definitiondf-10 9812 Define the number 10. See remarks under df-2 9804. (Contributed by NM, 5-Feb-2007.)

Theoremneg1cn 9813 -1 is a complex number. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1m1e0 9814 . Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2re 9815 The number 2 is real. (Contributed by NM, 27-May-1999.)

Theorem2cn 9816 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)

Theorem3re 9817 The number 3 is real. (Contributed by NM, 27-May-1999.)

Theorem3cn 9818 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)

Theorem4re 9819 The number 4 is real. (Contributed by NM, 27-May-1999.)

Theorem4cn 9820 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem5re 9821 The number 5 is real. (Contributed by NM, 27-May-1999.)

Theorem6re 9822 The number 6 is real. (Contributed by NM, 27-May-1999.)

Theorem7re 9823 The number 7 is real. (Contributed by NM, 27-May-1999.)

Theorem8re 9824 The number 8 is real. (Contributed by NM, 27-May-1999.)

Theorem9re 9825 The number 9 is real. (Contributed by NM, 27-May-1999.)

Theorem10re 9826 The number 10 is real. (Contributed by NM, 5-Feb-2007.)

Theorem0le0 9827 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2pos 9828 The number 2 is positive. (Contributed by NM, 27-May-1999.)

Theorem2ne0 9829 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)

Theorem3pos 9830 The number 3 is positive. (Contributed by NM, 27-May-1999.)

Theorem3ne0 9831 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorem4pos 9832 The number 4 is positive. (Contributed by NM, 27-May-1999.)

Theorem5pos 9833 The number 5 is positive. (Contributed by NM, 27-May-1999.)

Theorem6pos 9834 The number 6 is positive. (Contributed by NM, 27-May-1999.)

Theorem7pos 9835 The number 7 is positive. (Contributed by NM, 27-May-1999.)

Theorem8pos 9836 The number 8 is positive. (Contributed by NM, 27-May-1999.)

Theorem9pos 9837 The number 9 is positive. (Contributed by NM, 27-May-1999.)

Theorem10pos 9838 The number 10 is positive. (Contributed by NM, 5-Feb-2007.)

5.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

Theorem0p1e1 9839 Zero plus one equals one. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1p1e2 9840 One plus one equals two. (Contributed by NM, 1-Apr-2008.)

Theorem2m1e1 9841 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9859. (Contributed by David A. Wheeler, 4-Jan-2017.)

Theorem2p2e4 9842 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)

Theorem2times 9843 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremtimes2 9844 A number times 2. (Contributed by NM, 16-Oct-2007.)

Theorem2timesi 9845 Two times a number. (Contributed by NM, 1-Aug-1999.)

Theoremtimes2i 9846 A number times 2. (Contributed by NM, 11-May-2004.)

Theorem2p1e3 9847 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem3p1e4 9848 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem4p1e5 9849 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem5p1e6 9850 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem6p1e7 9851 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem7p1e8 9852 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem8p1e9 9853 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem9p1e10 9854 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem3p2e5 9855 3 + 2 = 5. (Contributed by NM, 11-May-2004.)

Theorem3p3e6 9856 3 + 3 = 6. (Contributed by NM, 11-May-2004.)

Theorem4p2e6 9857 4 + 2 = 6. (Contributed by NM, 11-May-2004.)

Theorem4p3e7 9858 4 + 3 = 7. (Contributed by NM, 11-May-2004.)

Theorem4p4e8 9859 4 + 4 = 8. (Contributed by NM, 11-May-2004.)

Theorem5p2e7 9860 5 + 2 = 7. (Contributed by NM, 11-May-2004.)

Theorem5p3e8 9861 5 + 3 = 8. (Contributed by NM, 11-May-2004.)

Theorem5p4e9 9862 5 + 4 = 9. (Contributed by NM, 11-May-2004.)

Theorem5p5e10 9863 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem6p2e8 9864 6 + 2 = 8. (Contributed by NM, 11-May-2004.)

Theorem6p3e9 9865 6 + 3 = 9. (Contributed by NM, 11-May-2004.)

Theorem6p4e10 9866 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem7p2e9 9867 7 + 2 = 9. (Contributed by NM, 11-May-2004.)

Theorem7p3e10 9868 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem8p2e10 9869 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem1t1e1 9870 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2t2e4 9871 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)

Theorem3t2e6 9872 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)

Theorem3t3e9 9873 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)

Theorem4t2e8 9874 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)

Theorem5t2e10 9875 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)

Theorem4d2e2 9876 One half of four is two. (Contributed by NM, 3-Sep-1999.)

Theorem2nn 9877 2 is a natural number. (Contributed by NM, 20-Aug-2001.)

Theorem3nn 9878 3 is a natural number. (Contributed by NM, 8-Jan-2006.)

Theorem4nn 9879 4 is a natural number. (Contributed by NM, 8-Jan-2006.)

Theorem5nn 9880 5 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem6nn 9881 6 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem7nn 9882 7 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem8nn 9883 8 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem9nn 9884 9 is a natural number. (Contributed by NM, 21-Oct-2012.)

Theorem10nn 9885 10 is a natural number. (Contributed by NM, 8-Nov-2012.)

Theorem1lt2 9886 1 is less than 2. (Contributed by NM, 24-Feb-2005.)

Theorem2lt3 9887 2 is less than 3. (Contributed by NM, 26-Sep-2010.)

Theorem1lt3 9888 1 is less than 3. (Contributed by NM, 26-Sep-2010.)

Theorem3lt4 9889 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt4 9890 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt4 9891 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem4lt5 9892 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem3lt5 9893 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt5 9894 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt5 9895 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem5lt6 9896 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem4lt6 9897 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem3lt6 9898 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt6 9899 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt6 9900 1 is less than 6. (Contributed by NM, 19-Oct-2012.)

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